I love book series which are set in the same world but about different characters or couples.
Is there a particular category or name for this type of series?
Any suggestions?

Some of my favorites are:

The Boy series by Meg Cabot
The Jane Austen diaries by Jenni James
The Brown siblings series and The Delicious series by Lauren Dane
The Dalvahini series by Lexi George
The Argeneau vampires by Lynsay Sands
The Love at stake series by Kerrelyn sparks
The Crowns' spies series by Julie Garwood

I need, need to recommend JR Ward's BDB series.

[brickablesmile]

I have no idea what the category is for these sort of series, but I would suggest, off the top of my head:

Celta's Heartmates series by Robin D. Owens
Dark series by Christine Feehan
Drake Sister series by Christine Feehan
Dark Hunter/Were-Hunter/Dream-Hunter series by Sherrilyn Kenyon

I feel that these books are similar to the Argeneau vampires by Lynsay Sands.

I've heard it referenced as a "shared world" series. But this also brings in other authors to write new characters and stories into the world so I'm not exactly sure if it fits in to a single author's usage.

[kaytja]

Perfect Chemistry by Simone Elkeles is really good

The Sartorias-deles books by Sherwood Smith. There's her 'historical' books, the Inda series and Banner of the Damned, which are more 'for older readers', and the rest of her books are set several hundred years after the historical books and are more YA.

John Norman's Gor series
Terry Pratchett's Discworld series

[RavenT]

The first shared world I read was Robert Lynne Aspirin's Thieves World anthologies:
http://en.wikipedia.org/wiki/Thieves%27_World

There were others like the Darkover anthologies around the same time.

The Wild Card series
Shadowrun series
most Mike Resnick and David Gemmell novels

[ficklefirefly]

You might enjoy the fantasy series The Hero and the Crown and The Blue Sword by Robin McKinley! The Blue Sword is the first book of the two, and it takes place well after (my hazy memory is saying centuries, actually) the events of The Hero and the Crown. Crown is one of my very favorite fantasy books from my youth! I have so much love for it. Sword, though, I came across quite a bit later than Crown, and didn't resonate as strongly.

Both were great, though, and well worth reading! I don't know whether to recommend reading Crown first (like I did) or Sword (which was published first).

ETA: I REALLY need to reread these.
One more edit: Here.

The first thing that comes to mind is books by Sarah Dessen. She writes exclusively young adult fiction; the books are set in the same area but all of her books are about different characters. Occasionally, a character from an earlier book will make a cameo or the characters will visit a popular bar or restaurant or nearby beach town. They also mention the same high schools. I think it's neat!